Theory and Calculation of Scattering with the Bethe-Salpeter Equation

The Bethe-Salpeter equation studied in this paper describes the interaction of two scalar particles via the exchange of a third scalar particle in the ladder approximation. The properties of the Green's function and the potential in coordinate space are shown to permit a Wick rotation to an imaginary time variable, without appeal to information not contained in the original equation. The resulting four-dimensional (Euclidean) wave equation has a solution which grows exponentially for large time-like distances but behaves as an ordinary Schr\"odinger scattering wave for large space-like distances. A modification of the Schwinger variational principle is used to obtain, with a modest use of computing machinery, scattering phase shifts for various angular momenta and for energies below the inelastic threshold. The success of these calculations indicates that the Bethe-Salpeter equation can be accepted as a powerful and practical tool for the study of strong-interaction dynamics.